Simple Divisibility Tests

overview

This page covers the following

• the divisibility test for 22

• the divisibility test for 1010

• the divisibility test for 33

• the divisibility test for 44

• the divisibility test for 55

• the divisibility test for 1111

• the divisibility test for 9

• the divisibility test for 6

Divisibility by 2

Consider the multiples of 2 given as 2,4,6,8,10,12,⋯. A similarity is observed in the multiples -- they are all even numbers.

Let us consider the numbers 4 and 5.

The even number 4 is divisible by 2. But the odd number 5 is not divisible by2.

Even numbers are divisible by 2. To check for divisibility by 2, only the digit in the units place is checked.

**Test for Divisibility by 2** : If the digit in the units place is even, then the number is divisible by 2.

Is 218 divisible by 2?

"Yes". Checking the digit in units place 8, it is concluded that the number is even and so divisible by 2.

Is 23 divisible by 2?

"No". Checking the digit in units place 3, it is concluded that the number is odd and so NOT divisible by 2.

Divisibility by 10

Consider the multiples of 10 given as 10,20,30,40,⋯. A similarity is observed in the multiples -- "they all have digit 0 in the units place". That is, all multiples of 10 end in 0.

Numbers with 0 in units place are divisible by 10.

Consider the numbers 20 and 24. Checking the units place, 20 has 0 and so 20 is divisible by 10. But 24 has 4 in units place, and so 24 is not divisible by 10.

**Test for Divisibility by 10** : If the digit in the units place is 0, then the number is divisible by 10.

Is 2008 divisible by 10?

"No". Checking the digit in units place 8, it is concluded that the number is not divisible by 10.

Is 920 divisible by 10?

"Yes". Checking the digit in units place 0, it is concluded that the number is divisible by 10.

Divisibility by 3

"3,6,9,12,15,18,21,24,27,30,33,⋯" are the multiples of 3. From the list, it is not easy to find a common property among the multiples.

Consider the multiples of 3 given as 3,6,9,12,15,18,21,24,27,30,33,⋯. The similarity is "sum of all digits is one of the numbers 3,6,9".

This is explained as follows

12→1+2=3 sum is 3

15→1+5=6 sum is 6

18→1+8=9 sum is 9

21→2+1=3 sum is 3

24→2+4=6 sum is 6

27→2+7=9 sum is 9

The property is true for all multiples of 3.

To identify the multiplies of 3, the sum of all digits of the number is checked for divisibility by 3.

*Explanation for the curious mind.*

Consider divisibility test 42 by 3

To simplify the divisibility test, let us subtract a multiple of the divisor 3.

Seeing the 10s place value 4, we choose the multiple 3×3×4=36 to subtract from the number.

As per the property of simplification by subtraction, the divisibility test of 42 is simplified into divisibility test of 42-36=40-36+2=4+2.

This explains the divisibility test for 3.

Consider the numbers 12 and 13.

12→1+2=3 sum 3 is divisible by 3, so 12 is divisible by 3.

13→1+3=4 sum 4 is not divisible by 3, so 13 is not divisible by 3.

**Test for Divisibility by 3** : If the sum of all the digits is divisible by 3 then the number is divisible by 3

Is 318 divisible by 3?

Yes. Checking the sum of digits 3+1+8=12. The sum is divisible by 3. It is concluded that the number is divisible by 3.

Is 923 divisible by 3?

No. Checking sum of the digits 9+2+3=14, it is concluded that the number is NOT divisible by 3.

Divisibility by 4

The multiples of 4 have the property that last two digits are divisible by 4.

*Explanation for the curious mind*

A large number like 2344 can be given as 23×100+44.

It was learned that if one addend is divisible, then the divisibility is decided by the other addend.

23×100 is always divisible by 4 as 100=25×4.

So, the divisibility test is done in 44, that is the tens and units digits.

**Test for Divisibility by 4** : If the last 2 digits (tens and units place value positions) of the number is divisible by 4 then the number is divisible by 4.

Is 2018 divisible by 4?

No. Checking the tens and units digits 18, it is concluded that the number is not divisible by 4.

Is 920 divisible by 4?

Yes. Checking the tens and units digits 20, it is concluded that the number is divisible by 4.

Divisibility by 5

the multiples of 5 are 5,10,15,20,25,⋯. The multiples of 5 has either 5 or 0 in the units place. The divisibility test for 5 is to check if the units digit is either 5 or 0.

*Explanation for the curious mind*

A large number like 2344 can be given as 234×10+4.

It was learned that if one addend is divisible, then the divisibility is decided by the other addend.

234×10 is divisible by 5 as 10=5×2.

So, the divisibility test is done on the last digit, that is the units digits. In the units digit the possible values are 5 and 0.

Consider 10 and 25. Both 10 and 25 are divisible by 5

10 has 0 in units place. So, 10 is divisible by 5.

25 has 5 in units place. So, 25 is divisible by 5.

**Test for Divisibility by 5** : If the units digit is 5 or 0, then the number is divisible by 5.

Is 2018 divisible by 5?

No. Checking the digit in units place 8, it is concluded that the number is not divisible by 5.

Is 920 divisible by 5?

Yes. Checking the digit in units place 0, it is concluded that the number is divisible by 5.

Divisibility by 11

Consider the multiples of 11 given as 11,22,33,⋯,110,121,132,⋯. Is there any similarity observed in the multiples?

The answer is "alternate digits have some pattern".

"Note: A number 3521 when multiplied by 11 is given as

3521×11

=3521×(10+1)

=35210+3521

=(3)(5+3)(2+5)(1+2)(1)

Four digit number, when multiplied by 11, results in

• units digit of product is same as that of multiplicand

• tens digit of product is sum of units and tens digit of multiplicand

• hundreds digit of product is sum of tens and hundreds digit of multiplicand

• and so on.

We can use this property to device a divisibility test for 11.

The multiples of 11 have the property explained below. For example, consider 121 (a multiple of 11).

Add the digits in odd positions 1+1, the result is 2

Add the digits in even positions 2.

Find the difference between these two results.

If the difference is 0 or a multiple of 11, then the number is a divisible by 11.

**Test for Divisibility by 11** : Find the sum of digits in even positions and the sum of digits in odd positions. If the difference between these two sums is a multiple of 11, then the number is divisible by 11.

Is 2108 divisible by 11?

No. The sum of alternate digits are 2+0=2 and 1+8=9. The difference between these sums are 9-2=7. Since 7 is not divisible by 11, the number is not divisible by 11.

Is 902 divisible by 11?

Yes. The 9+2-0=11, so the number is divisible by 11.

Divisibility by 9

Consider the multiples of 9 given as 9,18,27,36,⋯. A similarity is observed in these multiples : "sum of all digits is a multiple of 9".

This is explained as follows

18→1+8=9 sum is 9

27→2+7=9 sum is 9

36→3+6=9 sum is 9

189→1+8+9=18 sum is multiple of 9

This is true for all multiples of 9.

To identify the multiplies of 9, the sum of all digits of the number is checked for divisibility by 9.

*Explanation for the curious mind.*

Consider divisibility test 42 by 9

To simplify the divisibility test, let us subtract a multiple of the divisor 9.

Seeing the 10s place value 4, we choose the multiple 9×4 to subtract from the number.

As per the property of simplification by subtraction, the divisibility test of 42 is simplified into divisibility test of 42-36=40-36+2=4+2.

This explains the divisibility test for 9.

**Test for Divisibility by 9** : If the sum of all the digits is divisible by 9, then the number is divisible by 9

Is 2008 divisible by 9?

No. Checking the sum of digits 2+0+0+8=10, it is concluded that the number is not divisible by 9.

Is 927 divisible by 9?

Yes. Checking the sum of digits 9+2+7=18, it is concluded that the number is divisible by 9.

Divisibility by 6

Consider the number 3×2×2351, is it divisible by 6?

Yes, it is divisible by 6 as it has 3 and 2 as factors.

**Test for Divisibility by 6** : If the number is divisible by 2 and divisible by 3 then the number is divisible by 6

Is 2008 divisible by 6?

No. The number is divisible by 2 but not by 3. So it is not divisible by 6.

Is 912 divisible by 6?

Yes. The number is divisible by 2 and also divisible by 3. So it is divisible by 6.

summary

**Test for Divisibility by 2** : If the digit in the units place is even, then the number is divisible by 2.

**Test for Divisibility by 10** : If the digit in the units place is 0, then the number is divisible by 10.

**Test for Divisibility by 3** : If the sum of all the digits is divisible by 3 then the number is divisible by 3

**Test for Divisibility by 4** : If the last 2 digits (tens and units place value positions) of the number is divisible by 4 then the number is divisible by 4.

**Test for Divisibility by 5** : If the units digit is 5 or 0, then the number is divisible by 5.

**Test for Divisibility by 11** : Find the sum of digits in even positions and the sum of digits in odd positions. If the difference between these two sums is a multiple of 11, then the number is divisible by 11.

**Test for Divisibility by 9** : If the sum of all the digits is divisible by 9, then the number is divisible by 9

**Test for Divisibility by 6** : If the number is divisible by 2 and divisible by 3 then the number is divisible by 6

Outline

The outline of material to learn "Divisibility in Whole Numbers" is as follows.

Note: * click here for detailed outline of Whole divisors *

→ __Classification as odd, even, prime, and composite__

→ __Factors, Multiples, Prime factorization__

→ __Highest Common Factor__

→ __Lowest Common Multiple__

→ __Introduction to divisibility tests__

→ __Simple Divisibility Tests__

→ __Simplification of Divisibility Tests__

→ __Simplification in Digits for Divisibility Tests__